Singular surfaces, mod 2 homology, and hyperbolic volume, I
Authors:
Ian Agol, Marc Culler and Peter B. Shalen
Journal:
Trans. Amer. Math. Soc. 362 (2010), 34633498
MSC (2000):
Primary 57M50
Published electronically:
February 2, 2010
MathSciNet review:
2601597
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Additional Information
Abstract: If is a simple, closed, orientable manifold such that contains a genus surface group, and if has rank at least , we show that contains an embedded closed incompressible surface of genus at most . As an application we show that if is a closed orientable hyperbolic manifold of volume at most , then the rank of is at most .
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Jeff Weeks and Craig Hodgson, ClosedCensusInvariants.txt, downloadable text file.
 1.
 Ian Agol, Tameness of hyperbolic manifolds, arXiv:math/0405568v1[math.GT].
 2.
 Ian Agol, Peter A. Storm, and William P. Thurston, Lower bounds on volumes of hyperbolic Haken manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 10531077 (electronic). With an appendix by Nathan Dunfield. MR 2328715
 3.
 James W. Anderson, Richard D. Canary, Marc Culler, and Peter B. Shalen, Free Kleinian groups and volumes of hyperbolic manifolds, J. Differential Geom. 43 (1996), no. 4, 738782. MR 1412683 (98c:57012)
 4.
 Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385446 (electronic). MR 2188131 (2006g:57030)
 5.
 Marc Culler and Peter B. Shalen, Singular surfaces, mod homology, and hyperbolic volume, II, arXiv:math/070166v5[math.GT].
 6.
 Marc Culler and Peter B. Shalen, Paradoxical decompositions, generator Kleinian groups, and volumes of hyperbolic manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 231288. MR 1135928 (93a:57017)
 7.
 David Gabai, Foliations and the topology of manifolds, J. Differential Geom. 18 (1983), no. 3, 445503. MR 723813 (86a:57009)
 8.
 John Hempel, manifolds, AMS Chelsea Publishing, Providence, RI, 2004. Reprint of the 1976 original. MR 2098385 (2005e:57053)
 9.
 William Jaco and Peter B. Shalen, Peripheral structure of manifolds, Invent. Math. 38 (1976), no. 1, 5587. MR 0428332 (55:1357)
 10.
 William Meeks, III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621659. MR 678484 (84f:53053)
 11.
 C. D. Papakyriakopoulos, On Dehn's lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 126. MR 0090053 (19:761a)
 12.
 Grisha Perelman, Ricci flow with surgery on threemanifolds, arXiv:math/ 0303109v1[math.DG].
 13.
 Peter B. Shalen and Philip Wagreich, Growth rates, homology, and volumes of hyperbolic manifolds, Trans. Amer. Math. Soc. 331 (1992), no. 2, 895917. MR 1156298 (93d:57002)
 14.
 Arnold Shapiro and J. H. C. Whitehead, A proof and extension of Dehn's lemma, Bull. Amer. Math. Soc. 64 (1958), 174178. MR 0103474 (21:2242)
 15.
 Jonathan Simon, Compactification of covering spaces of compact manifolds, Michigan Math. J. 23 (1976), no. 3, 245256 (1977). MR 0431176 (55:4178)
 16.
 John Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 1219. MR 0121796 (22:12526)
 17.
 , Homology and central series of groups, J. Algebra 2 (1965), 170181. MR 0175956 (31:232)
 18.
 Friedhelm Waldhausen, On irreducible manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 5688. MR 0224099 (36:7146)
 19.
 Jeff Weeks and Craig Hodgson, ClosedCensusInvariants.txt, downloadable text file.
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Additional Information
Ian Agol
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, California 947203840
Email:
ianagol@math.berkeley.edu
Marc Culler
Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 606077045
Email:
culler@math.uic.edu
Peter B. Shalen
Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 606077045
Email:
shalen@math.uic.edu
DOI:
http://dx.doi.org/10.1090/S000299471004362X
PII:
S 00029947(10)04362X
Received by editor(s):
July 3, 2005
Received by editor(s) in revised form:
February 2, 2008
Published electronically:
February 2, 2010
Additional Notes:
This work was partially supported by NSF grants DMS0204142 and DMS0504975
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
